Write a unit vector making equal acute angles with the coordinate… - Vidyadip

Question

Write a unit vector making equal acute angles with the coordinates axes.

✓

Answer

Suppose $\vec{\text{r}}$ makes an angle $\alpha$ wuth each of the axis OX, OY and OZ.
Then, its direction cosines are $\text{l}=\cos\alpha,\ \text{m}=\cos\alpha,\ \text{n}=\cos\alpha$.
Now,
$\text{l}^2+\text{m}^2+\text{n}^2=1$
$\Rightarrow\ \text{l}^2+\text{l}^2+\text{l}^2=1$ $[\because\text{l = m = n}]$
$\Rightarrow\ 3\text{l}^2=1$
$\Rightarrow\ \text{l}^2=\frac{1}3$
$\Rightarrow\ \text{l}=\pm\frac{1}{\sqrt3}$
Since the angle is acute Hence, we take only positive value
Therefore, unit vector is $\Big(\frac{1}{\sqrt3}\hat{\text{i}}+\frac{1}{\sqrt3}\hat{\text{j}}+\frac{1}{\sqrt3}\hat{\text{k}}\Big)$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "2 Marks" from Algebra of Vectors→Algebra of Vectors — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\vec{\text{a}}$ and $\vec{\text{b}}$ are unit vectors such that $\vec{\text{a}}\times\vec{\text{b}}$ is also a unit vector, find the angle between $\vec{\text{a}}$ and $\vec{\text{b}}.$

There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?

Compute the products AB and BA whichever exists the following cases:
$\text{A}=\begin{bmatrix}1&-1&2&3\end{bmatrix}$ and $\text{B}=\begin{bmatrix}0\\1\\3\\2\end{bmatrix}$

Evaluate the following integrals:

$\int\frac{\sec^2\text{x}}{\sqrt{4+\tan^2\text{x}}}\text{ dx}$

Evaluate the following integrals:
$\int\limits^{1}_02^{\text{x}-[\text{x}]}\text{dx}$

If u, v and w are functions of x, then show that
$\frac{d}{d x}(u . v . w)=\frac{d u}{d x} v . w+u . \frac{d v}{d x} \cdot w+u \cdot v \frac{d w}{d x}$
in two ways - first by repeated application of product rule, second by logarithmic differentiation.

Write minors and cofactors of the elements of $\left|\begin{array}{cc}2 & -4 \\ 0 & 3\end{array}\right|$.

Compute the products AB and BA whichever exists the following cases:
$\text{A}=\begin{bmatrix}1&-2\\2&3\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&2&3\\2&3&1\end{bmatrix}$

Integrate the function: $\frac{{\cos \sqrt x }}{{\sqrt x }}$

Let * be a binary operation on the set Q of rational numbers as follows:
$\text{a} * \text{b} = \frac{\text{ab}}{4}$